Alexander and Jackson — Polygons to (venerate Diagrams^ S^-c. 7 



On each diagram the slope of the shearing force locus is 42 tons vertical 

 to 21 feet horizontal, or 2 to 1. On fig. 7 the downward loads add 17 tons to 

 the shear (support) at the left end. The locus slopes in five parts, which are 

 produced to meet the base. The first slope meets the base at half of 59 feet 

 from the left end, or at 8-|- feet past the centre. This, then, is the highest 

 point of the bending-moment diagram for the first 5-foot interval of the 

 span, and is therefore the centre of the first arc up to JS. The centres for 

 the other arcs are Jialf of 5, 8, 10, and 7, nearer the left end, one after 

 another. 



On the other diagram, fig. 8, the shear at the left end of the girder begins 

 with 25 tons, being 17 less than 42, so that now the first slope meets the 

 base 12^ feet from the left end, or 8^ feet to the left of the centre. The other 

 four slopes meet the base at points further from the left end by hcdf of 

 5, 8, 10, and 7, and so give the centres for the arcs, as shown on the diagram 

 below. 



On fig. 7 the locus of shearing force crosses the base once, so that the 

 height of D is the mas. of maxima. Its value is Jilo = 630 ft.-tous, so that 

 the height of Z) should scale off at -/'fiSO = 25'1 on the scale, common to 

 feet and sq. roots of foot-tons. 



But on fig. 8 the locus of shearing crosses the base three times : at the 

 centre, at 2 feet left, and at 3 feet right of the centre of the span. The height 

 of i> is a min. of maxima, lying between the two super-maxima, whose values 

 are given by the heights of a^ = 16 and a.^ = 161, these being the radii of the 

 arcs CI) and BE. Hence M^ = 16- = 256 and M, = 261, while the height of D 

 squared gives 261 - 3^ or 256 - 2^ = 252 ft.-tons. 



On the two loci of figs. 7 and 8, formed by the ares of semicircles inter- 

 lacing at B, 0, D, and E, the average of the squares of the heights, at a pair of 

 corresponding points, equals the square of the height to a semicircle standing 

 on the span. 



If the four intermediate loads be halved, the shearing force locus shows 

 that the distances apart of the centres of the semicircles are halved also. As 

 the loads are further and further decreased, the semicircles come closer and 

 closer together ; and when the four loads are altogether removed, they all 

 coincide with the semicircle standing on the span. It is the diagram of 

 square roots of the bending moments due to the spread load of 84 tons 

 alone. 



Consider the locomotive. If its wheels come closer and closer together 

 proportionately, the centres of the semicircles do the same, and the modified 

 locus suits the shortened locomotive. If the loco, be further shortened till 



